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Two-Scale Estimates for Special Finite Discrete Operators.

1 Introduction

We consider the following Calderon-Zygmund singular integral operator [10]

[mathematical expression not reproducible](1.1)

in the Lebesgue space [L.sub.2] (h[Z.sup.m]) and its discrete analogue of the following type

[mathematical expression not reproducible], (1.2)

in the space [L.sub.2] (h[R.sup.m]) [equivalent to] [l.sup.2] of functions [u.sub.d] of a discrete variable [mathematical expression not reproducible].

We recall [10] that that a symbol [sigma]([xi]) of the operator K is the Fourier transform of its kernel in principal value sense

[mathematical expression not reproducible]

and such symbol is called an elliptic symbol if

[mathematical expression not reproducible].

Our main goal is the following. Starting from the operator (1.1) we introduce the discrete operator (1.2) acting in infinite dimensional space so that it preserves basic properties of the operator (1.1) related to an ellipticity and invertibility [18,19,20,21,22]. Further to obtain computational algorithms we would like to construct finite dimensional analogue of the operator (1.2) preserving same properties, and to obtain results on comparison of these operators and solutions of corresponding equations.

1.1 Some previous approaches and studies

In books [2,3,5,6,14,15,17] authors present studies on convolution equations, one-dimensional singular integral equations and multidimensional weakly singular integral equations related to approximate solution of these equations.

As usual there are the following questions in these studies: 1) to find an approximate equation desirable in a finite-dimensional (N-dimensional) space so that under enough large N this approximate equation will be uniquely solvable in an appropriate space; 2) to obtain an error estimate between exact solution of an initial equation and exact solution of approximate equation in dependence on N. As far as we know these questions were investigated fully for one-dimensional singular integral equations on smooth curves L in a complex plane C [8,15]

[mathematical expression not reproducible]

convolution equations with integrable kernel K(x) on a straight line

[mathematical expression not reproducible]

and some their multidimensional analogues [2, 3, 5] and for multidimensional integral equations with a weak singularity

[mathematical expression not reproducible]

where D is a domain with a smooth boundary [partial derivative]D and the kernel K(x, y) satisfies the estimate

[mathematical expression not reproducible]

last equations are generated by compact integral operators in appropriate spaces [17].

An algebraic approach based on a local principle for studying certain finite approximations for integral operators was used in many papers, and its development for last years is presented in books for example [2,3,6]. But this method of [C.sup.*]-algebras permits to prove a solvability of approximating equation but it can't help for obtaining an error estimate for solutions of these equations. Moreover concrete applications of the method are related as a rule to one-dimensional singular integral operators and equations.

For an error estimate there are some computational results [9,12,13,16], and a lot of results are related to special equations for applied problems (see for example [7]). Multidimensional case is more complicated because there is no such advanced theory similar classical Riemann boundary value problem [4,11].

We would like to note that in general there are no valuable results for approximate solution of multidimensional singular integral equations with Calderon --Zygmund operators although such equations in distinct form arise in many problems of partial differential equations and mathematical physics [10]. In our opinion a direct digitization is more convenient for computer calculations than other methods, and we try to use and justify this approach.

2 Discrete spaces and transformations

2.1 Definitions and notations

We accept the following conventions and notations. The kernel K(x) is a differentiate function on the unit sphere in [R.sup.m], K(0) [equivalent to] 0, and [K.sub.d] is a restriction of the kernel K on lattice points h[Z.sup.m]. Let h be a size of a "spatial quant", N be a size of our "Universe". These parameters will be tend to zero and infinity respectively. For the operator (1.1) we'll use the following reduction. First we replace the operator (1.1) by the discrete operator (1.2), and second we approximate this series by a special finite sum

[mathematical expression not reproducible] (2.1)

For [K.sub.d,N] in the formula (2.1) we suggest a following construction. If [K.sub.d] is a restriction of the continual kernel K on lattice points h[Z.sup.m] then we take a restriction of the [K.sub.d] on points h[Z.sup.m] [intersection] [Q.sub.N], where

[mathematical expression not reproducible]

and denote by [K.sub.d,N] its periodic continuation on the whole h[Z.sup.m].

We would like to justify a following sequence of transformations, "continual" operator (1.1) [right arrow] "infinite discrete" operator (1.2) [right arrow] "finite discrete" operator (2.1) with corresponding estimates on h and N. The comparison of (1.1) and (1.2) was given in papers' series [18,19,20,21,22], and here we consider a comparison between (1.2) and (2.1).

Let's denote by PN the restriction operator h[Z.sup.m] [right arrow] h[Z.sup.m] [intersection] [Q.sub.N], and the space [L.sub.2] (h[Z.sup.m] [intersection] [Q.sub.N]) is denoted by [l.sup.2.sub.N] so that [P.sub.N] is a projector [l.sup.2] [right arrow] [l.sup.2.sub.N].

Definition 1. Approximation rate of operators [K.sub.d] and [K.sub.d,N] is called the following operator norm

[mathematical expression not reproducible]

It is non-trivial to obtain an estimate for the operator norm, but we'll give an estimate for an individual element assuming the existence of some its properties. More precisely we'll suppose that the element ud is a restriction of the function u which has Holder property in [R.sup.m] and [mathematical expression not reproducible] with some [gamma] > 0.

We will obtain a "weak estimate" for approximation rate but enough for our purposes. We assume additionally that a function [u.sub.d] is a restriction on h[Z.sup.m] of continuous function with certain estimates [20,21]. Let's define the discrete space [C.sub.h]([alpha], [beta]) as a functional space of discrete variable [mathematical expression not reproducible] with finite norm

[mathematical expression not reproducible]

It means that the function [u.sub.d] [member of] [C.sub.h]([alpha], [beta]) satisfies the following estimates

[mathematical expression not reproducible]

Let us note that under required assumptions [C.sub.h]([alpha], [beta]) [subset] [L.sub.2](h[Z.sup.m]), and these discrete space are discrete analogue of corresponding subspaces of continuous functions [1].

Theorem 1. For operators [K.sub.d] and [K.sub.d,N] the following estimate

[mathematical expression not reproducible]

holds, where constant C doesn't depend on N, h.

Proof. Let us write

[mathematical expression not reproducible]

where I is an identity operator in [L.sup.2](h[Z.sup.m]).

First two summands have annihilated, and we need to estimate only the last summand. We have

[mathematical expression not reproducible]

because norms of operators [K.sub.d] are uniformly bounded, and for the last norm taking into account above estimates we can write

[mathematical expression not reproducible]

and further

[mathematical expression not reproducible]

The last integral using spherical coordinates gives the estimate [N.sup.m+2([alpha]-[beta]) which tends to 0 under n [right arrow] [delta] if [beta] > [alpha] + m/2.

Remark 1. Similar theorem was obtained in [19,20] for the space [C.sub.h]([alpha], [beta]).

This theorem plays a key role for obtaining an estimate for approximate solution of an multidimensional singular integral equation with the operator (1.1) and permits to use fast Fourier transform for evaluating a numerical solution. Some test calculations were given in [20].

2.2 Discrete Fourier transform

We define the discrete Fourier transform for a function ud of a discrete variable [mathematical expression not reproducible] as the series

[mathematical expression not reproducible],

where [mathematical expression not reproducible].

This discrete Fourier transform has same properties like standard continual Fourier transform, particularly for a discrete convolution of two discrete functions [u.sub.d], [u.sub.d]

[mathematical expression not reproducible]

we have the well known multiplication property

([F.sub.d]([u.sub.d] * [v.sub.d]))([xi]) = ([F.sub.d][u.sub.d])([xi]) * ([F.sub.d][v.sub.d])(e).

If we apply this property to the operator [K.sub.d] we obtain

([F.sub.d]([K.sub.d][u.sub.d]))([xi]) = ([F.sub.d][K.sub.d])([xi]) * ([F.sub.d][u.sub.d])([xi]).

Let us denote ([F.sub.d][K.sub.d])([xi]) [equivalent to] [[sigma].sub.d]([xi]) and give the following

Definition 2. The function [mathematical expression not reproducible], is called a periodic symbol of the operator [K.sub.d].

We will assume below that the symbol [[sigma].sub.d]([xi]) [member of] C([??][T.sup.m]) therefore we have immediately the following

Property 1. The operator [K.sub.d] is invertible in the space [L.sub.2](h[Z.sub.m]) iff [mathematical expression not reproducible].

Definition 3. A continuous periodic symbol is called an elliptic symbol if [mathematical expression not reproducible].

So we see that an arbitrary elliptic periodic symbol [[sigma].sub.d]([xi]) corresponds to an invertible operator [K.sub.d] in the space [L.sub.2](h[Z.sup.m]).

Remark 2. It was proved earlier that operators (1.1) and (1.2) for cases D = [R.sup.m], D = [R.sup.m.sub.t] are invertible or non-invertible in spaces [L.sub.2]([R.sup.m]), [L.sub.2]([R.sup.m.sub.+]) and [L.sub.2](h[Z.sup.m]), [L.sub.2](h[Z.sup.m.sub.+]) simultaneously [18,22].

2.3 Finite discrete Fourier transform

Definition 4. For a function [u.sub.d,N] [member of] [l.sup.2.sub.N] its finite discrete Fourier transform is defined by the formula

[mathematical expression not reproducible]

Definition 5. A symbol of the operator [K.sub.d,N] is called the function [[sigma].sub.d,N] ([xi]) of a discrete variable [mathematical expression not reproducible] defined by the formula

[mathematical expression not reproducible]

3 Comparison between infinite and finite discrete operators

Theorem 2. If the operator [K.sub.d] is invertible in the space [l.sup.2] then the operator [K.sub.d,N] is invertible in the space [l.sup.2.sub.N] for enough large N.

Proof. Let the function

[mathematical expression not reproducible]

is a segment of the Fourier series

[mathematical expression not reproducible]

and according our assumptions this is continuous function on [??][T.sup.m]. Therefore values of the partial sum coincide with values of [[sigma].sub.d,N] in points [mathematical expression not reproducible]. Besides these partial sums are continuous functions on [??][T.sup.m].

3.1 Some auxiliary results

Lemma 1. The norm of operator [K.sub.d]; [l.sup.2] [right arrow] [l.sup.2] doesn't depend on h

For the proof see [19].

Lemma 2. The norm of operator [K.sup.d,N] : [l.sup.2.sub.N] [right arrow] [l.sup.2.sub.N] doesn't depend on N, h.

Proof. Using the Theorem 2 and the property that the norm of the operator [K.sub.d,N] is equivalent to max [mathematical expression not reproducible] (see also [10]) we obtain the required assertion.

4 Correlation between N, h and location of [xi]

The proved Theorem 1 is a very rough estimate. It is possible to obtain more exact error estimate for finite discrete solution taking into account a location of the point [??] and relations between parameters N and h.

We consider three types of equations

K u = v, (4.1)

[K.sub.d] [u.sub.d] = [Pd.sup.v] [equivalent to] [v.sub.d], (4.2)

where [P.sub.d] is a restriction operator which given continuous function v defined on [R.sup.m] maps to a collection of its values on [Z.sup.m], and

[K.sub.d,N] [u.sub.d,N] = [v.sub.d,N] [equivalent to] [P.sub.N] vd (4.3)

and would like to have an estimate for nearness of their solutions.

Let us denote by r([??]) the distance between [??] [member of] h[Z.sup.m] [intersection] [Q.sub.N].

Theorem 3. Let [mathematical expression not reproducible] the following estimate

[mathematical expression not reproducible]

holds, [c.sub.1], [c.sub.2] are constants non-depending on h, N.

Proof. Assuming N is enough large so that both operators [K.sub.d] and [K.sub.d,N] are invertible in spaces [l.sup.2] and [l.sup.2,N] respectively let us consider the difference

[mathematical expression not reproducible]

where

[mathematical expression not reproducible]

We'll consider summands separately and give pointwise estimates for them. For

[mathematical expression not reproducible]

we have [I.sub.2]([??]) = 0 because this difference is not zero for points from h[Z.sup.m]\[Q.sub.N] only, so we need to estimate [I.sub.1]([??]) only.

First we need to say some words on a structure of operators [K.sup.-1.sub.d] and [K.sup.-1.sub.d,N] which we have constructed for operators [mathematical expression not reproducible]. Lemma 1 [19] implies that the norm of the operator [K.sub.d] doesn't depend on h. Also we have an analogue assertion for the operator [K.sub.d,N] (Lemma 2). Moreover the operator [K.sup.-1.sub.d] is generated by Calderon-Zygmund operator with symbol [[sigma].sup.-1]([xi]) and corresponding kernel [K.sup.-1](x), so that the kernel [K.sup.-1.sub.2]([??]) of the discrete operator [K.sup.-1.sub.2] is a restriction of the kernel [K.sup.- 1](x) on the lattice h[Z.sup.m].

Further the operator [K.sup.-1.sub.dN] is constructed by the same way. We take the discrete kernel [K.sup.-1.sub.d]([??]) then we take its restriction on [Q.sub.N] and a periodic continuation on a whole h[Z.sup.m]. A symbol of a such operator will be[mathematical expression not reproducible], and we conserve all required properties.

Let us start to estimate [I.sub.1] ([??]).

[mathematical expression not reproducible]

and we need to estimate the last sum only.

1) [mathematical expression not reproducible]. Here taking into account that [mathematical expression not reproducible] we obtain

[mathematical expression not reproducible].

We'll represent [I.sub.1]([??]) as a sum [I.sub.1]([??]) = [I.sub.11]([??]) + [I.sub.12]([??]), where

[mathematical expression not reproducible]

and [mathematical expression not reproducible] we assume that [mathematical expression not reproducible]. Then

[mathematical expression not reproducible]

and for enough small h we have

[mathematical expression not reproducible]

where [D.sub.N] ([??]) is a ball with a center in x and radius ~ N. Using spherical coordinates with a center in [??] we obtain

[mathematical expression not reproducible]

and thus

[mathematical expression not reproducible]

For [I.sub.12] ([??]) we use another estimate. We write

consequently [mathematical expression not reproducible], and then

[mathematical expression not reproducible]. Thus

[mathematical expression not reproducible]

Since we are interested in small h the last sum can be dominated by the following integral

[mathematical expression not reproducible]

and calculations with spherical coordinates give the estimate for [beta] - [alpha] > 0

[mathematical expression not reproducible]

3) r([??]) ~ 1. So we have the following estimate

[mathematical expression not reproducible]

because [mathematical expression not reproducible]. Since we are interested in small h the last sum can be dominated by the following integral

[mathematical expression not reproducible]

and calculations with spherical coordinates give the estimate for [beta] - [alpha] > 0

[mathematical expression not reproducible]

5 Conclusions

Our considerations give a certain algorithm for solving a simplest singular integral equation in a whole space [R.sup.m], and also in [R.sup.m.sub.+] p taking into account authors' conclusions in [21,22]. The error estimate for finite discrete solutions shows that varying N, h we can obtain a necessary sharpness. Collecting all authors' results [18,19,20,21,22] related to equations (4.1), (4.2), (4.3) we conclude that there is a certain correspondence between solvability of these equations and their solutions.

https://doi.org/10.3846/13926292.2017.1307790

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Alexander V. Vasilyev (a,b) and Vladimir B. Vasilyev (a,b)

(a) Belgorod National Research University Studencheskaya 14/1, 308007 Belgorod, Russia

(b) Lipetsk State Technical University

E-mail(corresp.): vladimir.b.vasilyev@gmail.com

E-mail: alexvassel@gmail.com

Received October 9, 2016; revised March 4, 2017; published online May 15, 2017

* This work was supported by RFBR and Lipetsk regional government of Russia, grant No. 14-41-03595-r-center-a
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